Time-dependent density functional theory

Max-Planck Institute of Microstructure Physics

Halle (Saale)

E.K.U. Gross

OUTLINE • Phenomena to be described by TDDFT: Matter in weak and strong laser fields • Basic theorems of TDDFT

• Approximate xc functionals of TDDFT the “exact adiabatic” approximation • TDDFT in the linear-response regime

-- Optical excitation spectra of molecules -- Excitonic effects in the optical spectra of solids -- Charge-transfer excitations in molecules and the discontinuity of the xc kernel

THANKS

Manfred Lein Volker Engel Erich Runge Stephan Kümmel M. Thiele Martin Petersilka Ulrich Gossmann Tobias Grabo Sangeeta Sharma Kay Dewhurst Antonio Sanna Maria Hellgren

Time-dependent systems

Weak laser (vlaser(t) << ven) : Calculate 1. Linear density response ρ1(r t) 2. Dynamical polarizability 3. Photo-absorption cross section

→

( ) ( )∫ ωρ−=ωα rd,r zEe 3

1

( ) απω

−=ωσ Imc

4

Strong laser (vlaser(t) ≥ ven) : Non-perturbative solution of full TDSE required

Generic situation: Molecule in laser field

∑α

++=, j

eee W T H(t) + E ·rj·sin ωt → → Zα e2

|rj-Rα|

Weak lasers: Optical spectra of

finite systems (atoms, molecules)

continuum states

unoccupied bound states

occupied bound states

I1

I2

Laser frequency ω

σ(ω)

phot

o-ab

sorp

tion

cros

s sec

tion

Standard linear response formalism

H(t0) = full static Hamiltonian at t0

full response function

( )0 mH t m E m= ← exact many-body eigenfunctions and energies of system

⇒ The exact linear density response has poles at the exact excitation energies Ω = Em - E0

ρ1 (ω) = χ (ω) v1

( ) ( ) ( )( )

( ) ( )( )m0 m 0 m 0

ˆ ˆ ˆ ˆ0 r m m r 0 0 r ' m m r ' 0r, r ';

E E i E E ilim

+η→

ρ ρ ρ ρχ ω = − ω − − + η ω + − + η

∑

Weak lasers: Optical spectra of

periodic solids (insulators)

Strong Laser Fields

Comparison: Electric field on 1st Bohr-orbit in hydrogen Intensities in the range of 1013 …1016 W/cm2

92

0 0

1 eE 5.1 10 V m4 a

= = ×πε

2 16 20

1I cE 3.51 10 W cm2

= ε = ×

e-

a0 +

Three quantities to look at: I. Emitted ions II. Emitted electrons III. Emitted photons

Walker et al., PRL 73, 1227 (1994)

λ = 780 nm

I. Emitted Ions

Momentum Distribution of the He2+ recoil ions

(M. Lein, E.K.U.G., V. Engel, J. Phys. B 33, 433 (2000))

( ) 21 2p , p , tΨ of the He atom

( ) 21 2p , p , tΨ of the He atom

(M. Lein, E.K.U.G., V. Engel, J. Phys. B 33, 433 (2000))

(M. Lein, E.K.U.G., V. Engel, PRL 85, 4707 (2000))

vLaser(z,t) = E z sin ωt I = 1015 W/cm2 λ = 780 nm

Wigner distribution W(Z,P,t) of the electronic center of mass for He atom

II. Electrons: Above-Threshold-Ionization (ATI)

Ionized electrons absorb more photons than necessary to overcome the ionization potential (IP)

Photoelectrons: ( )kinE n s IP= + ω −⇒ Equidistant maxima in intervals of : ω

Agostini et al., PRL 42, 1127 (1979)

IP

ω

III. Photons: High-Harmonic Generation

Emission of photons whose frequencies are integer multiples of the driving field. Over a wide frequency range, the peak intensities are almost constant (plateau).

log(

Inte

nsity

)

ω

pIP 3.17U+

ωωωωω

5 ω

Even harmonic generation due to nuclear motion

T. Kreibich, M. Lein, V. Engel, E.K.U.G., PRL 87, 103901 (2001)

(a) Harmonic spectrum generated from the model HD molecule driven by a laser with peak intensity 1014 W/cm2 and wavelength 770 nm. The plotted quantity is proportional to the number of emitted phonons. (b) Same as panel (a) for the model H2 molecule.

HD

H2

Molecular Electronics

left lead L central region C right lead R

Dream: Use single molecules as basic units (transistors, diodes, …) of electronic devices

Molecular Electronics

left lead L central region C right lead R

Bias between L and R is turned on: U(t) V for large t

A steady current, I, may develop as a result.

• Calculate current-voltage characteristics I(V) • Investigate cases where no steady state is achieved

Dream: Use single molecules as basic units (transistors, diodes, …) of electronic devices

Hamiltonian for the complete system of Ne electrons with coordinates (r1 ··· rNe) ≡ r and Nn nuclei with coordinates (R1 ··· RNn) ≡ R, masses M1 ··· MNn and charges Z1 ··· ZNn.

n nn

e ee en

ˆ ˆ ˆH T (R) W (R)ˆ ˆ ˆT (r) W (r) U (R, r)

= +

+ + +

with en n

e e n

N2N N2i

n e nn1 i 1 ,

N N N

ee enj,k j 1 1j k jj k

Z Z1ˆ ˆ ˆT T W2M 2m 2 R R

Z1 1ˆ ˆW U2 r r r R

µ νν

ν= = µ νν µ νµ≠ν

ν

= ν= ν≠

∇ ∇= − = − =

−

= = −− −

∑ ∑ ∑

∑ ∑∑

Time-dependent Schrödinger equation

( ) ( ) ( )( ) ( )externali r, R, t H r,R V r,R, t r,R, tt

∂Ψ = + ψ

∂

Example: Oxygen atom (8 electrons)

depends on 24 coordinates

rough table of the wavefunction

10 entries per coordinate: ⇒ 1024 entries 1 byte per entry: ⇒ 1024 bytes 1010 bytes per DVD: ⇒ 1014 DVDs 10 g per DVD: ⇒ 1015 g DVDs = 109 t DVDs

( )81 r,,r

Ψ

Why don’t we just solve the many-particle SE?

ESSENCE OF DENSITY-FUNTIONAL THEORY

• Every observable quantity of a quantum system can be calculated from the density of the system ALONE

• The density of particles interacting with each other can be calculated as the density of an auxiliary system of non-interacting particles

Ground-State Density Functional Theory

compare ground-state densities ρ(r) resulting from different external potentials v(r).

QUESTION: Are the ground-state densities coming from different potentials always different?

ρ(r)

v(r)

Ground-State Density Functional Theory

v(r) Ψ (r1…rN) ρ (r)

single-particle potentials having nondegenerate ground state

ground-state wavefunctions

ground-state densities

Hohenberg-Kohn-Theorem (1964)

G: v(r) → ρ (r) is invertible

A

G

Ã

Ground-State Density Functional Theory

For any observable O

[ ] [ ] [ ]ˆ ˆO O OΨ Ψ = Ψ ρ Ψ ρ = ρ

is functional of the density

For any observable O

[ ] [ ] [ ]ˆ ˆO O OΨ Ψ = Ψ ρ Ψ ρ = ρ

is functional of the density

For general observables: Need to find approximations for the functionals O[ρ] representing the observables.

For any observable O

[ ] [ ] [ ]ˆ ˆO O OΨ Ψ = Ψ ρ Ψ ρ = ρ

is functional of the density

For general observables: Need to find approximations for the functionals O[ρ] representing the observables. Easy observables: Etot[ρ], D[ρ] = ʃ ρ(r) r d3r, … More difficult: spectroscopic information, gaps, ….

ESSENCE OF DENSITY-FUNTIONAL THEORY

• Every observable quantity of a quantum system can be calculated from the density of the system ALONE (Hohenberg, Kohn, 1964)

• The density of particles interacting with each other can be calculated as the density of an auxiliary system of non-interacting particles

(Kohn, Sham, 1965)

[ ]( )extv ρ r ( )ρ r [ ]( )sv ρ r

HK 1-1 mapping for interacting particles

HK 1-1 mapping for non-interacting particles

Kohn-Sham Theorem

Let ρo(r) be the ground-state density of interacting electrons moving in the external potential vo(r). Then there exists a local potential vs,o(r) such that non-interacting particles exposed to vs,o(r) have the ground-state density ρo(r), i.e.

( ) ( ) ( )2

s,o j j jv r r r2

∇− + ϕ =∈ ϕ

( ) ( )

j

N 2

o jj (with lowest )

ρ r r∈

= ϕ∑

proof: Note: The KS equations do not follow from the variational principle!!

Uniqueness follows from HK 1-1 mapping Existence follows from V-representability theorem

( ) [ ]( )s,o s ov r v ρ r=

,

By construction, the HK mapping is well-defined for all those functions ρ(r) that are ground-state densities of some potential (so called V-representable functions ρ(r)).

QUESTION: Are all “reasonable” functions ρ(r) V-representable?

V-representability theorem (Chayes, Chayes, Ruskai, J Stat. Phys. 38, 497 (1985))

On a lattice (finite or infinite), any normalizable positive function ρ(r), that is compatible with the Pauli principle, is (both interacting and non-interacting) ensemble-V-representable.

In other words: For any given ρ(r) (normalizable, positive, compatible with Pauli principle) there exists a potential, vext[ρ](r), yielding ρ(r) as interacting ground-state density, and there exists another potential, vs[ρ](r), yielding ρ(r) as non-interacting ground-state density.

In the worst case, the potential has degenerate ground states such that the given ρ(r) is representable as a linear combination of the degenerate ground-state densities (ensemble-V-representable).

Basic 1-1 correspondence: The time-dependent density determines uniquely the time-dependent external potential and hence all physical observables for fixed initial state.

( ) ( )v rt rt1-1←→ ρ

Time-dependent density-functional formalism

KS theorem: The time-dependent density of the interacting system of interest can be calculated as density

of an auxiliary non-interacting (KS) system with the local potential

( ) ( )2N

j = 1 ρ rt = rtjϕ∑

( ) ( ) ( ) ( )3S

r ' tv r ' t ' rt v rt d r '

r r 'ρ

ρ = + + −∫

( ) [ ]( ) ( )2 2

j S ji rt v rt rtt 2m

∂ ∇ϕ = − + ρ ϕ ∂

( ) ( )xcv r ' t ' rt ρ

(E. Runge, E.K.U.G., PRL 52, 997 (1984))

define maps ( ) ( )F : v r t t Ψ

( ) ( )F : t r t Ψ ρ

densities

( )r tρ

wave functions

( )tΨ

potentials

( )v r tF

˜ F solve tdSE with fixed

Ψ t o( ) = Ψo

( )( ) ( ) ( )

r t

ˆt r t

ρ =

Ψ ρ Ψ

( ) ( ) ( )s ss

ˆ ˆ ˆr r r+ρ = ψ ψ∑

G

( ) ( )G : v r t r t ρ

Proof of basic 1-1 correspondence between and ( )v r t ( )ρ r t

THEOREM (time-dependent analogue of Hohenberg-Kohn theorem)

The map

( ) ( )G : v r t r t ρ

defined for all single-particle potentials which can be expanded into a Taylor series with respect to the time coordinate around to

is invertible up to within an additive merely time-dependent function in the potential.

( )v r t

Proof: to be shown: ( )r tρ

( )v ' r t( )v r t

cannot happen

i.e. ( ) ( ) ( ) ( ) ( )ˆ ˆv r t v ' r t c t r t ' r t ≠ + ⇒ ρ ≠ ρ

potential expandable into Taylor series

( ) ( )j r t j ' r t≠

( ) ( )r t ' r tρ ≠ ρ

step 1

step 2

!

( ) ( )o

k

k t tk 0 v r t v ' r t constant

t :

=

∂ ∃ ≥ − ≠ ∂

i. the basic 1-1 mapping for interacting and non-interacting particles

ii. the TD V-representability theorem (R. van Leeuwen, PRL 82, 3863 (1999)).

The TDKS equations follow (like in the static case) from:

A TDDFT variational principle exists as well, but this is more tricky: R. van Leeuwen S. Mukamel T. Gal G. Vignale . .

Simplest possible approximation for [ ]( )xcv ρ r t

Adiabatic Approximation:

statxcvwhere = xc potential of ground-state DFT

( )adiab statxc xc

n ( r 't )v r t : v [n](r)

=ρ=

Simplest possible approximation for [ ]( )xcv ρ r t

Adiabatic Approximation:

statxcvwhere = xc potential of ground-state DFT

( )adiab statxc xc

n ( r 't )v r t : v [n](r)

=ρ=

• Every approximate ground-state xc potential yields an adiabatic approximation to the time-dependent xc potential • Deficiencies of approximate ground-state xc potentials are inherited by the corresponding adiabatic functionals ( ) ( )ALDA hom

xc xcn ( r t )

Example : v r t : v n=ρ

=

has the wrong asymptotic form for large r

Two sources of troubles: • deficiciences of ground-state xc functional

• adiabatic approximation

Two sources of troubles: • deficiciences of ground-state xc functional

• adiabatic approximation

Which problem is more severe? Investigate the best possible adiabatic approximation, i.e. the one that follows from the exact ground-state xc potential

• Solve 1D model for He atom in strong laser fields (numerically) exactly. This yields exact TD density ρ(r,t).

• Inversion of one-particle TDSE yields exact TDKS potential. Then, subtracting the laser field and the TD-Hartree term, yields the exact TD xc potential .

• Inversion of one-particle ground-state SE yields the exact static KS potential, vKS-static[ρ(t)], that gives (for each separate t) ρ(r,t) as ground-state density.

• Inversion of the many-particle ground-state SE yields the static external potential, vext-static[ρ(t)], that gives (for each separate t) ρ(r,t) as interacting ground-state density .

• Compare the exact TD xc potential of step 1 with the exact adiabatic approximation which is obtained by subtraction :

vxc-exact-adiab(t) = vKS-static[ρ(t)] – vH[ρ(t)] – vext-static[ρ(t)]

Solid line: exact Dashed line: exact adiabatic

E(t) ramped over 27 a.u. (0.65 fs) to the value E=0.14 a.u. and then kept constant

t = 0 t = 21.5 a.u. t = 43 a.u.

M. Thiele, E.K.U.G., S. Kuemmel, Phys. Rev. Lett. 100, 153004 (2008)

M. Thiele, E.K.U.G., S. Kuemmel, Phys. Rev. Lett. 100, 153004 (2008)

4-cycle pulse with λ = 780 nm, I1= 4x1014W/cm2, I2=7x1014W/cm2

Solid line: exact Dashed line: exact adiabatic

LINEAR RESPONSE THEORY

t = t0 : Interacting system in ground state of potential v0(r) with density ρ0(r) t > t0 : Switch on perturbation v1(r t) (with v1(r t0)=0). Density: ρ(r t) = ρ0(r) + δρ(r t)

Consider functional ρ[v](r t) defined by solution of interacting TDSE

Functional Taylor expansion of ρ[v] around vo:

[ ] ( ) [ ] ( )0ρ v rt ρ v rt = + 1v

[ ] ( )( )

3δρ v rt d r'dt'

δv r't'0v

+∫ ( )1v r ' t '

[ ] ( )0ρ v rt=

[ ] ( )( ) ( )

23 3δ ρ v rt1 d r'd r"dt'dt"

2 δv r't' δv r"t"0v

+ ∫ ∫ ( ) ( )1 1v r ', t ' v r", t " ( )2ρ rt

( )1ρ rt

( )oρ r

…

ρ1(r,t) = linear density response of interacting system

( ) [ ] ( )( )

δρ v rtrt, r ' t ' :

δv r't'0v

χ = = density-density response function of interacting system

Analogous function ρs[vs](r t) for non-interacting system

[ ] ( ) ( ) ( ) [ ] ( )( )

S S 3S S S S,0 S S,0

S

δρ v rtρ v rt ρ v rt ρ v rt d r'dt'

δv r't'S,0v

= + = + + ∫ S,1v ( )S,1v r't'

( ) [ ] ( )( )

S SS

S

δρ v rtrt, r ' t ' :

δv r't'S,0v

χ = = density-density response function of non-interacting system

Standard linear response formalism

H(t0) = full static Hamiltonian at t0

full response function

( )0 mH t m E m= ← exact many-body eigenfunctions and energies of system

⇒ The exact linear density response has poles at the exact excitation energies Ω = Em - E0

ρ1 (ω) = χ (ω) v1

( ) ( ) ( )( )

( ) ( )( )m0 m 0 m 0

ˆ ˆ ˆ ˆ0 r m m r 0 0 r ' m m r ' 0r, r ';

E E i E E ilim

+η→

ρ ρ ρ ρχ ω = − ω − − + η ω + − + η

∑

GOAL: Find a way to calculate ρ1(r t) without explicitly evaluating χ(r t,r't') of the interacting system

starting point: Definition of xc potential

[ ]( ) [ ]( ) [ ]( ) [ ]( )xc S ext Hv ρ rt : v ρ rt v ρ rt v ρ rt= − −

Notes: • vxc is well-defined through non-interacting / interacting 1-1 mapping.

• vS[ρ] depends on initial determinant Φ0. • vext[ρ] depends on initial many-body state Ψ0.

⇒ In general, only if system is initially in ground-state then, via HK, Φ0

and Ψ0 are determined by ρ0 and vxc depends on ρ alone.

[ ]xc xc 0 0v v ρ,Φ ,Ψ =

[ ]( )( )

[ ]( )( )

[ ]( )( )

( )xc S extδv ρ rt δv ρ rt δv ρ rt δ t t'δρ r't' δρ r't' δρ r't' r r'

0 0 0ρ ρ ρ

−= − −

−

( )xcf rt, r't' ( )1S rt, r't'−χ ( )1 rt, r't'−χ ( )eeV rt, r't'

[ ]( )( )

[ ]( )( )

[ ]( )( )

( )xc S extδv ρ rt δv ρ rt δv ρ rt δ t t'δρ r't' δρ r't' δρ r't' r r'

0 0 0ρ ρ ρ

−= − −

−

1 1xc ee Sf V − −+ = χ − χ

( )xcf rt, r't' ( )1S rt, r't'−χ ( )1 rt, r't'−χ ( )eeV rt, r't'

[ ]( )( )

[ ]( )( )

[ ]( )( )

( )xc S extδv ρ rt δv ρ rt δv ρ rt δ t t'δρ r't' δρ r't' δρ r't' r r'

0 0 0ρ ρ ρ

−= − −

−

1 1xc ee Sf V − −+ = χ − χ • χ χS •

( )xcf rt, r't' ( )1S rt, r't'−χ ( )1 rt, r't'−χ ( )CW rt, r't'

[ ]( )( )

[ ]( )( )

[ ]( )( )

( )xc S extδv ρ rt δv ρ rt δv ρ rt δ t t'δρ r't' δρ r't' δρ r't' r r'

0 0 0ρ ρ ρ

−= − −

−

( )S xc ee Sχ f V χ χ χ+ = −

1 1xc ee Sf V − −+ = χ − χ • χ χS •

( )xcf rt, r't' ( )1S rt, r't'−χ ( )1 rt, r't'−χ ( )CW rt, r't'

[ ]( )( )

[ ]( )( )

[ ]( )( )

( )xc S extδv ρ rt δv ρ rt δv ρ rt δ t t'δρ r't' δρ r't' δρ r't' r r'

0 0 0ρ ρ ρ

−= − −

−

( )S S ee xcχ χ χ V f χ= + +

Act with this operator equation on arbitrary v1(r t) and use χ v1 = ρ1 :

• Exact integral equation for ρ1(r t), to be solved iteratively

• Need approximation for

(either for fxc directly or for vxc)

( ) ( ) ( ) ( ) ( ) ( )3 31 S 1 ee xc 1ρ rt = d r'dt'χ rt, r't' v rt + d r"dt" V r't', r"t" + f r't', r"t" ρ r"t"

∫ ∫

( ) [ ]( )( )

xcxc

δv ρ r't'f r't', r"t"

δρ r"t"0ρ

=

Adiabatic approximation

[ ] ( ) [ ] ( )adiab static DFTxc xcv ρ rt := v rt( )ρ t

e.g. adiabatic LDA: ( ) ( )ALDA LDAxc xcv rt : v α= = −( )ρ rt ( )1 3ρ rt +

( ) ( )( ) ( ) ( ) ( ) ( )

ALDA ALDAxcALDA xc

xc

δv rt v f rt, r't' δ r r' δ t t'δρ r't' ρ r

0 0ρ ρ r

∂⇒ = = − −

∂

( ) ( )( )

2 homxc2

eδ r r' δ t t'n

0ρ r

∂= − −

∂

In the adiabatic approximation, the xc potential vxc(t) at time t only depends on the density ρ(t) at the very same point in time.

Total photoabsorption cross section of the Xe atom versus photon energy in the vicinity of the 4d threshold.

Solid line: self-consistent time-dependent KS calculation [A. Zangwill and P. Soven, PRA 21, 1561 (1980)]; crosses: experimental data [R. Haensel, G. Keitel, P. Schreiber, and C. Kunz, Phys. Rev. 188, 1375 (1969)].

Photo-absorption in weak lasers

continuum states

unoccupied bound states

occupied bound states

I1

I2

No absorption if ω < lowest excitation energy

Laser frequency ω

σ(ω)

phot

o-ab

sorp

tion

cros

s sec

tion

Discrete excitation energies from TDDFT

exact representation of linear density response:

( ) ( ) ( ) ( ) ( ) ( )( )1 S 1 ee 1 xc 1ˆˆˆ v V fρ ω = χ ω ω + ρ ω + ω ρ ω

“^” denotes integral operators, i.e. ( ) ( ) 3xc 1 xc 1f f r, r ' r ' d r 'ρ ≡ ρ∫

where

with

( ) ( )( )

jkS

j,k j k

M r, r 'ˆ r, r ';

iχ ω =

ω − ε − ε + η∑

( ) ( ) ( ) ( ) ( ) ( )* *jk k j j j k kM r, r ' f f r r ' r ' r= − ϕ ϕ ϕ ϕ

mm

m

1 if f

0 if

ϕ= ϕ

j kε − ε KS excitation energy

is occupied in KS ground state

is unoccupied in KS ground state

( ) ( )( ) ( ) ( ) ( )S ee xc 1 S 1ˆˆ ˆˆ ˆ1 V f v − χ ω + ω ρ ω = χ ω ω

ρ1(ω) → ∞ for ω → Ω (exact excitation energy) but right-hand side remains finite for ω → Ω

hence ( ) ( )( ) ( ) ( ) ( )S ee xcˆˆ ˆˆ1 V f − χ ω + ω ξ ω = λ ω ξ ω

λ(ω) → 0 for ω → Ω This condition rigorously determines the exact excitation energies, i.e.,

( ) ( )( ) ( )S ee xcˆˆ ˆˆ1 V f 0 − χ Ω + Ω ξ Ω =

This leads to the (non-linear) eigenvalue equation (See T. Grabo, M. Petersilka, E. K. U. G., J. Mol. Struc. (Theochem) 501, 353 (2000))

( )( )qq ' q qq ' q ' qq '

M Ω + ω δ β = Ωβ∑

( ) ( ) ( )3 3qq' q' q xc q'

1M α d r d r'Φ r f r, r',Ω Φ r'r r'

= + −

∫ ∫

( )q j,a=

( ) ( ) ( )*q a jr r rΦ = ϕ ϕ

q a jf fα = −

jaq ε−ε=ω

double index

Equivalent to Casida equations if: • ω-dependence of fxc is neglected • orbitals are real-valued • pure density-response is considered in Casida eqs.

AtomExperimental Excitation

Energies 1S→1P(in Ry)

KS energydifferences∆∈KS (Ry)

∆∈KS + K

Be 0.388 0.259 0.391Mg 0.319 0.234 0.327Ca 0.216 0.157 0.234

Zn 0.426 0.315 0.423

Sr 0.198 0.141 0.210Cd 0.398 0.269 0.391

from: M. Petersilka, U. J. Gossmann, E.K.U.G., PRL 76, 1212 (1996)

∆E = ∆∈KS + K ∈j - ∈k

( ) ( ) ( ) ( ) ( )3 3j j k k xc

1K d r d r ' r r ' r ' r f r, r 'r r '

∗ ∗ = ϕ ϕ ϕ ϕ + −

∫ ∫

Excitation energies of CO molecule

State Ωexpt KS-transition ∆∈KS ∆∈KS + KA 1Π 0.3127 5Σ→2Π 0.2523 0.3267

a 3Π 0.2323 0.2238

I 1Σ - 0.3631 1Π→2Π 0.3626 0.3626

D 1∆ 0.3759 0.3812

a' 3Σ+ 0.3127 0.3181

e 3Σ - 0.3631 0.3626

d 3∆ 0.3440 0.3404

approximations made: vxc and fxc LDA ALDA

T. Grabo, M. Petersilka and E.K.U. Gross, J. Mol. Struct. (Theochem) 501, 353 (2000)

Quantum defects in Helium

Bare KS exact x-ALDA xc-ALDA (VWN) x-TDOEP

( )[ ]n 2

n

1E a.u.2 n

= −− µ

M. Petersilka, U.J. Gossmann and E.K.U.G., in: Electronic Density Functional Theory: Recent Progress and New Directions, J.F. Dobson, G. Vignale, M.P. Das, ed(s), (Plenum, New York, 1998), p 177 - 197.

(M. Petersilka, E.K.U.G., K. Burke, Int. J. Quantum Chem. 80, 534 (2000))

Failures of ALDA in the linear response regime

• response of long chains strongly overestimated (see: Champagne et al., JCP 109, 10489 (1998) and 110, 11664 (1999))

• in the optical spectrum of periodic solids, there are two problems:

--absorption edge is wrong (LDA KS gap) --excitons are missing

• charge-transfer excitations not properly described (see: Dreuw et al., JCP 119, 2943 (2003))

• H2 dissociation is incorrect:

(see: Gritsenko, van Gisbergen, Görling, Baerends, JCP 113, 8478 (2000))

( ) ( )1 1u g RE E 0 + +

→∞Σ − Σ → (in ALDA)

ALDA

Solid Argon

L. Reining, V. Olevano, A. Rubio, G. Onida, PRL 88, 066404 (2002)

OBSERVATION: In the long-wavelength-limit (q = 0), relevent for optical absorption, ALDA is not reliable. In particular, excitonic peaks are completely missing. Results are very close to RPA.

EXPLANATION: In the TDDFT response equation, the bare Coulomb interaction and the xc kernel only appear as sum (Vee + fxc). For q 0, Vee diverges like 1/q2, while fxc in ALDA goes to a constant. Hence results are close to fxc = 0 (RPA) in the q 0 limit.

CONCLUSION: exact property Approximations for fxc are needed which, for q 0, correctly diverge like 1/q2. Such approximations can be derived from many-body perturbation theory (see, e.g., L. Reining, V. Olevano, A. Rubio, G. Onida, PRL 88, 066404 (2002)).

2q 0f 1 qexact

xc →→

Excitons in TDDFT

( ) ( ) ( ) ( )( ) ( )11

S xc S( , ) 1 , 1 , ,−− ε ω = + χ ω − + ω χ ω q q q q q qv v f

( ) ( ) ( )xc xc, , t t ' , t , t '− ≡ δ δρr r' r r'f v

( ) ( ) ( ) ( ) ( ) 11RPA S S, 1 , 1 ,

−−ε ω = + χ ω − χ ω q q q q qv v

xc 0≡f

xc kernel

RPA corresponds to:

TDDFT response equation (a matrix equation)

( )S ,χ ωq Kohn-Sham response function

exact equation

( ) ( ) ( )( )

( )( )

1

00 00RPA

bootxc

1

S

,, 0, 0

01 , 0

,−−ε ω =

= − =ε ω

ε ω =ω

ω= − χ =

q qq q

qqf

v

( ) ( ) ( ) ( ) ( )( ) ( )appS

rS

1 1

xcf1 , 1 ,,,−−

ε ω = + χ ω − + χ ωω qqq q q qv v

Bootstrap kernel (Sharma, Dewhurst, Sanna, EKUG, PRL 107, 186401 (2011))

( ) ( ) ( ) ( )S 0

bootx

prc cap

x1, ,

,0

, 10

= − +χ ω = χ

ωωω =q q

q qf f

where χ0 is a single-particle response function that has the right gap, e.g. G0W0 , LDA+U, LDA+Scissors, Tran-Blaha functional, …

Charge transfer excitations and the discontinuity of fxc

A B

charge-transfer excitation

In exact DFT IP(A) = – εHOMO EA(B) = – εLUMO – ∆xc

(A)

(B)

derivative discontinuity

CT excitation energy

( ) ( ) ( ) ( )∫ ∫ −

ϕϕ−−≈Ω

'rr'r r

'rdrdEAIP2

B

2

A33BACT

( ) ( ) ( ) ( )∫ ∫ −

ϕϕ−ε−ε≈Ω

'rr'r r

'rdrd2

B

2

A33AMOL

BMOLCT

R

(A) Infinite R

= – εMOL = – εMOL

(B)

Finite (but large) R

(B)

( ) ( ) ( ) ( )∫ ∫ −

ϕϕ−ε−ε≈Ω

'rr'r r

'rdrd2

B

2

A33AMOL

BMOLCT

~ 1/R

( ) ( ) ( ) ( )∫ ∫ −

ϕϕ−ε−ε≈Ω

'rr'r r

'rdrd2

B

2

A33AMOL

BMOLCT

~ 1/R

( ) ( ) ∫ ∫ ϕϕΩϕϕ−ε−ε≈Ω )r()r(),r'(r, )f(r')(r''rdrd *BACTxcB

*A

33AMOL

BMOLCT

In TDDFT (single-pole approximation)

( ) ( ) ( ) ( )∫ ∫ −

ϕϕ−ε−ε≈Ω

'rr'r r

'rdrd2

B

2

A33AMOL

BMOLCT

~ 1/R

( ) ( ) ∫ ∫ ϕϕΩϕϕ−ε−ε≈Ω )r()r(),r'(r, )f(r')(r''rdrd *BACTxcB

*A

33AMOL

BMOLCT

In TDDFT (single-pole approximation)

Exponentially small

Exponentially small

( ) ( ) ( ) ( )∫ ∫ −

ϕϕ−ε−ε≈Ω

'rr'r r

'rdrd2

B

2

A33AMOL

BMOLCT

~ 1/R

( ) ( ) ∫ ∫ ϕϕΩϕϕ−ε−ε≈Ω )r()r(),r'(r, )f(r')(r''rdrd *BACTxcB

*A

33AMOL

BMOLCT

In TDDFT (single-pole approximation)

Exponentially small

Exponentially small

CONCLUSIONS: To describe CT excitations correctly • vxc must have proper derivative discontinuities • fxc(r,r’) must increase exponentially as function of r and of r’ for ω→ΩCT

( ) ( ) ( ) ( )xc xc xc xc, , g g+ −′ ′ ′= + +r r r r r rf f

( )xc xc xcv ( ) v+ −= + ∆r r

Discontinuity of vxc(r):

Discontinuity of fxc(r,r’):

Δxc is constant throughout space

M. Hellgren, EKUG, PRA 85, 022514 (2012)

( ) ( )( )

( )s2 2A I rLxcg ~ ~ e

n− −ϕ r

rr

r → ∞

This exponentially increasing behavour is achieved by the discontinuity:

M. Hellgren, EKUG, PRA 85, 022514 (2012)

He-Be neutral diatomic in 1D

M. Hellgren, EKUG, PRA 85, 022514 (2012)

( )CT q x q q x2 q v q 1 Rω = ω + + ω → ω + ∆ −f

He-Be neutral diatomic in 1D

SFB 450 SFB 658 SPP 1145 SFB 762